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The distributive property is an essential tool in algebra that helps us simplify expressions. It states that for any numbers or variables a, b, and c: \[ a(b + c) = ab + ac \] This means you multiply the number outside the parentheses by each term inside the parentheses. Let's see how this works with an example from our exercise.
For part (a), we had to distribute 4 to the terms inside the parentheses: \[ 4(6 - \sqrt{11}) = (4 \cdot 6) - (4 \cdot \sqrt{11}) \] Breaking it down, you see that 4 times 6 is 24 and 4 times \sqrt{11} is just rewritten as 4\sqrt{11}. So, we get: \[ 24 - 4\sqrt{11} \] The distributive property helps by removing the parentheses, making the expression easier to work with.

Simplifying square roots involves breaking down the square root into smaller, more manageable parts. This useful technique helps simplify expressions, making them easier to comprehend and work with.
For part (b) of our exercise, we started with the expression \sqrt{2}(5 - \sqrt{12}). Here, we can simplify the square root of 12: \[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \] Substituting this back into the expression gives us: \[ \sqrt{2}(5 - 2\sqrt{3}) \] With the simpler square root value, our algebraic manipulation becomes easier.

Algebraic expressions involve variables, constants, and arithmetic operations. Understanding how to simplify these expressions is crucial in algebra.
In our exercise, simplifying algebraic expressions requires a combination of techniques. We've seen how distributing and simplifying square roots are key steps.
For the expression \sqrt{2}(5 - 2\sqrt{3}), we use both these techniques. Distributing \sqrt{2} to both terms inside the parentheses, we get: \[ \sqrt{2}(5 - 2\sqrt{3}) = (\sqrt{2} \cdot 5) - (\sqrt{2} \cdot 2\sqrt{3}) = 5\sqrt{2} - 2\sqrt{6} \] This final simplified form is easier to interpret and helps solve further algebraic problems.
By mastering these fundamental methods, you can simplify complex algebraic expressions confidently and efficiently.